340 JACOBI'S METHOD ^ f [220,

Next we perform with the function ^ all the operations which
have been performed with the function <f> ; then the desired
common integral

j;=a2

will be obtained, except in the single case when we have

(/„ *)=*!=''>

where c' is a determinate constant.

From a combination of these respective exceptional cases,
which are the only ones in each of which the common integral
F^ has not been obtained, we can construct a common integral

.F2. For let

be substituted in (F, Jra) = 0 = (/1, jPs); then these equations
become

Now the former equation is satisfied identically since <£ and
are both integrals of the subsidiary equations (A) ; while since

!.' (/,

;{{ , and (

'4\*

>jj the latter equation becomes

I ' ' '

I j< ' This is satisfied by

and therefore jP2 = @ (c'^> - c&) = a2,
where ® is any arbitrary functional symbol (which may at will be
chosen of a simple form), is the desired integral.
Hence in every case the common integral of the equations.
which determine F^ has been found; for convenience we may
denote it by